1 research outputs found
A Multilevel Approach for Trace System in HDG Discretizations
We propose a multilevel approach for trace systems resulting from hybridized
discontinuous Galerkin (HDG) methods. The key is to blend ideas from nested
dissection, domain decomposition, and high-order characteristic of HDG
discretizations. Specifically, we first create a coarse solver by eliminating
and/or limiting the front growth in nested dissection. This is accomplished by
projecting the trace data into a sequence of same or high-order polynomials on
a set of increasingly coarser edges/faces. We then combine the coarse
solver with a block-Jacobi fine scale solver to form a two-level
solver/preconditioner. Numerical experiments indicate that the performance of
the resulting two-level solver/preconditioner depends only on the smoothness of
the solution and is independent of the nature of the PDE under consideration.
While the proposed algorithms are developed within the HDG framework, they are
applicable to other hybrid(ized) high-order finite element methods. Moreover,
we show that our multilevel algorithms can be interpreted as a multigrid method
with specific intergrid transfer and smoothing operators. With several
numerical examples from Poisson, pure transport, and convection-diffusion
equations we demonstrate the robustness and scalability of the algorithms